Question

Evaluate each expression exactly. $$\csc \left[\cos ^{-1}\left(\frac{1}{4}\right)\right]$$

   Evaluate each expression exactly.
$$\csc \left[\cos ^{-1}\left(\frac{1}{4}\right)\right]$$

Precalculus

Cynthia Young 2nd Edition

Chapter 6, Problem 85

Instant Answer

Solved by Expert Shamshad Waris

06/25/2020

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This means that $\cos(\theta) = \frac{1}{4}$. Show more…

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Evaluate each expression exactly. $$\csc \left[\cos ^{-1}\left(\frac{1}{4}\right)\right]$$

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Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, such as the cosecant, are defined as the reciprocals of the sine function. Understanding that csc? = 1/sin? is fundamental to computing the exact value of composite expressions involving trig functions, especially when combined with inverse functions and identities.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to determine the angle whose cosine value is known. When you see an expression like cos?¹(1/4), it signifies finding the angle whose cosine is 1/4, which is critical for understanding how to evaluate composite trigonometric expressions.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is essential for deriving one trigonometric ratio when another is known. In the context of the problem, given cos?, the identity allows for the computation of sin?, enabling the evaluation of trigonometric expressions that involve functions like cosecant.

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